According to Wikipedia (and common knowledge):
Where natural levels of iodine in the soil are low and the iodine is
not taken up by vegetables, iodine added to salt provides the small
but essential amount of iodine needed by humans.
However, this article from Harvard tells us to pay attention when consuming iodized salt along with "hidden salt".
Iodized salt can theoretically provide all the required daily recommended iodine quantity, but this will lead to consumption of a rather great quantity of salt:
To get all your iodine from salt, you would need more than half a
teaspoon of iodized salt a day. That's two-thirds of the daily
allotment of sodium (1,500 milligrams) recommended by the American
In the mean time:
between 75% and 90% of sodium in the average American's diet comes
from prepared or processed food, and most food companies don't use
So, it is recommended to obtain iodine mostly from food, if possible:
[...] you can cut back on salt and not worry about losing out on this
important element. Ocean-caught or ocean-farmed fish and shellfish
tend to be naturally rich in iodine. Other good sources include milk,
cheese, yogurt, eggs, and vegetables grown in iodine-rich soil.
Multivitamin pills that also contain minerals usually provide 150
micrograms of iodine.
Conclusion: iodized salt and plants in common doses should be fine, but multivitamin pills can be considered when in doubt about reaching the required iodine recommended quantity.